Efficient computation of the<i>W</i><sub>3</sub>topological invariant and application to Floquet–Bloch systems

We introduce Z2-valued bulk invariants for symmetry-protected topological phases in 2 + 1 dimensional driven quantum systems. These invariants adapt the W3-invariant, expressed as a sum over degeneracy points of the propagator, to the respective symmetry class of the Floquet-Bloch Hamiltonian. The bulk-boundary correspondence that holds for each invariant relates a non-zero value of the bulk invariant to the existence of symmetry-protected topological boundary states. To demonstrate this correspondence we apply our invariants to a chiral Harper, time-reversal KaneMele, and particle-hole symmetric graphene model with periodic driving, where they successfully predict the appearance of boundary states that exist despite the trivial topological character of the Floquet bands. Especially for particle-hole symmetry, combination of the W3 and the Z2-invariants allows us to distinguish between weak and strong topological phases.

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“…Particularly efficient computation of the Z 2 -invariants is possible with the algorithm from Ref. 30.…”

Section: Discussionmentioning

confidence: 99%

“…(4.4) in Ref. 30, which is the basis of the algorithm presented there. Owing to this similarity, evaluation of the above expression, and also of the Z 2 -invariants defined in the main text, is possible with that algorithm.…”

Section: Acknowledgmentsmentioning

confidence: 99%

Topological invariants for Floquet-Bloch systems with chiral, time-reversal, or particle-hole symmetry

2018

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“…over all degeneracy points i = 1, ..., dp of the Floquet-Bloch propagator U (k, t) that occur during time-evolution [28,30]. To each degeneracy point, we assign a topological chargeĈ i , given as a Chern number, and a weight factor N i (ε) that ensures that only the degeneracy points in the gap ε contribute to the sum.…”

Section: Ii43 Winding Number Wmentioning

confidence: 99%

Fermionic time-reversal symmetry in a photonic topological insulator

et al. 2020

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Much of the recent enthusiasm directed towards topological insulators [1][2][3][4][5][6][7][8][9][10][11][12][13] as a new state of matter is motivated by their hallmark feature of protected chiral edge states. In fermionic systems, Kramers degeneracy gives rise to these entities in the presence of time-reversal symmetry (TRS) [1-3, 14, 15]. In contrast, bosonic systems obeying TRS are generally assumed to be fundamentally precluded from supporting edge states [3,16]. In this work, we dispel this perception and experimentally demonstrate counterpropagating chiral states at the edge of a time-reversal-symmetric photonic waveguide structure. The pivotal step in our approach is encoding the effective spin of the propagating states as a degree of freedom of the underlying waveguide lattice, such that our photonic topological insulator is characterised by a Z 2 -type invariant. Our findings allow for fermionic properties to be harnessed in bosonic systems, thereby opening new avenues for topological physics in photonics as well as acoustics, mechanics and even matter waves. arXiv:1812.07930v1 [physics.optics]

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“…[32], computed with the algorithm from Ref. [34]). This confirms that the driving protocol indeed supports a nontrivial time-reversal symmetric topological phase, with a pair of counterpropagating boundary states.…”

Section: A Time-reversal Symmetrymentioning

confidence: 99%

Universal driving protocol for symmetry-protected Floquet topological phases

2019

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We propose a universal driving protocol for the realization of symmetry-protected topological phases in 2 + 1 dimensional Floquet systems. Our proposal is based on the theoretical analysis of the possible symmetries of a square lattice model with pairwise nearest-neighbor coupling terms. Among the eight possible symmetry operators we identify the two relevant choices for topological phases with either time-reversal, chiral, or particle-hole symmetry. From the corresponding symmetry conditions on the protocol parameters, we obtain the universal driving protocol where each of the symmetries can be realized or broken individually. We provide specific parameter values for the different cases, and demonstrate the existence of symmetry-protected copropagating and counterpropagating topological boundary states. The driving protocol especially allows us to switch between bosonic and fermionic time-reversal symmetry, and thus between a trivial and non-trivial symmetry-protected topological phase, through continuous variation of a parameter. TRS CS PHS Π 2 = 1 step 1 ∆A = ∆B = ∆ ∆A = ∆ ∆C = ∆D = −∆ ∆C = −∆ this J = 2π/T J = 2π/T J = 2π/T work ∆ = 3/T ∆ = 9/T J = π/T

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Efficient computation of theW₃topological invariant and application to Floquet–Bloch systems

Cited by 14 publications

References 39 publications

Topological invariants for Floquet-Bloch systems with chiral, time-reversal, or particle-hole symmetry

Topological invariants for Floquet-Bloch systems with chiral, time-reversal, or particle-hole symmetry

Fermionic time-reversal symmetry in a photonic topological insulator

Universal driving protocol for symmetry-protected Floquet topological phases

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Efficient computation of theW3topological invariant and application to Floquet–Bloch systems

Cited by 14 publications

References 39 publications

Topological invariants for Floquet-Bloch systems with chiral, time-reversal, or particle-hole symmetry

Topological invariants for Floquet-Bloch systems with chiral, time-reversal, or particle-hole symmetry

Fermionic time-reversal symmetry in a photonic topological insulator

Universal driving protocol for symmetry-protected Floquet topological phases

Contact Info

Product

Resources

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Efficient computation of theW₃topological invariant and application to Floquet–Bloch systems